Rainbow Restrained Domination Numbers in Graphs

Abstract

A \(2\)-rainbow dominating function (2RDF) of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V(G)\) with \(f(v) = \emptyset\), the condition that there exists \(u \in N(v)\) with \(\bigcup_{u\in N(v)}f(u) = \{1,2\}\) is fulfilled, where \(N(v)\) is the open neighborhood of \(v\). A rainbow dominating function \(f\) is said to be a rainbow restrained domination function if the induced subgraph of \(G\) by the vertices with label \(\emptyset\) has no isolated vertex. The weight of a rainbow restrained dominating function is the value \(w(f) = \sum_{u \in V(G)} |f(u)|\). The minimum weight of a rainbow restrained dominating function of \(G\) is called the rainbow restrained domination number of \(G\). In this paper, we initiate the study of the rainbow restrained domination number and we present some bounds for this parameter.